A Spectral notion of Gromov-Wasserstein distances

We introduce a spectral notion of distance between shapes (closed Riemannian manifolds) and study its theoretical properties. We show that our distance satisfies the properties of a metric on the class of isometric shapes, which means, in particular, that two shapes are at 0 distance if and only if they are isometric. Our construction is similar to the recently proposed Gromov-Wasserstein distance, but rather than viewing shapes merely as metric spaces, we define our distance via the comparison of heat kernels. This allows us to relate our distance to previously proposed spectral invariants used for shape comparison, such as the spectrum of the Laplace-Beltrami operator and statistics of diffusion distances. In addition, the heat kernel provides a natural notion of scale, which is useful for multi-scale shape comparison. We also prove a hierarchy of lower bounds for our distance, which provide increasing discriminative power at the cost of increase in computational complexity.
http://www.math.yale.edu/~tl292/seminar/appliedanalysis